MASAS in Type $II_{1}$ factor

Question

Let $N$ be a type $II_{1}$ factor. Does there exist a diffuse abelian sub algebra of $N$ which is not Maximal abelian?

Answer 1

In general, a diffuse abelian subalgebra of $N$ may not be a masa of $N$. Choose a projection $p$ in $N$ with trace $1/2$. Take a masa $M$ in $pNp$. There exists a unitary operator $u$ in $N$ such that $u^*pu=p^{\perp}$. Then, $\{x+u^*xu:x\in M\}$ is diffuse and abelian but not maximal.